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pacman::p_load(sf, tidyverse, funModeling, blorr, corrplot,
ggpubr, spdep, GWmodel, tmap, skimr, caret)In-class Exercise 5
Overview
This is an In-class Exercise that focuses on geographically weighted logistic regression.
Getting Started
Loading the Packages
Importing the Data
Importing the analytical data
Using read_rds() we import the analytical data into the R environment.
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osun <- read_rds("data/Osun.rds")
osun_wp_sf <- read_rds("data/Osun_wp_sf.rds")We then produce the following simple feature objects:
- `โosunโ (โOsun.rdsโ) - the polygon data containing the boundaries of Osun
- โosun_wp_sfโ (โOsun_wp_sf.rdsโ ) - the data of water points found in Osun
Exploratory Data Analysis
Distribution of 'status'
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osun_wp_sf %>% freq(input = 'status')
status frequency percentage cumulative_perc
1 TRUE 2642 55.5 55.5
2 FALSE 2118 44.5 100.0We do this to check our dependent variable. Since later on we will be doing logistic regression, we need to make sure our data isnโt biased to one outcome.
Summary Statistics with skimr
The function skim() provides a comprehensive summary of our variables as shown below.
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osun_wp_sf %>% skim()| Name | Piped data |
| Number of rows | 4760 |
| Number of columns | 75 |
| _______________________ | |
| Column type frequency: | |
| character | 47 |
| logical | 5 |
| numeric | 23 |
| ________________________ | |
| Group variables | None |
Variable type: character
| skim_variable | n_missing | complete_rate | min | max | empty | n_unique | whitespace |
|---|---|---|---|---|---|---|---|
| source | 0 | 1.00 | 5 | 44 | 0 | 2 | 0 |
| report_date | 0 | 1.00 | 22 | 22 | 0 | 42 | 0 |
| status_id | 0 | 1.00 | 2 | 7 | 0 | 3 | 0 |
| water_source_clean | 0 | 1.00 | 8 | 22 | 0 | 3 | 0 |
| water_source_category | 0 | 1.00 | 4 | 6 | 0 | 2 | 0 |
| water_tech_clean | 24 | 0.99 | 9 | 23 | 0 | 3 | 0 |
| water_tech_category | 24 | 0.99 | 9 | 15 | 0 | 2 | 0 |
| facility_type | 0 | 1.00 | 8 | 8 | 0 | 1 | 0 |
| clean_country_name | 0 | 1.00 | 7 | 7 | 0 | 1 | 0 |
| clean_adm1 | 0 | 1.00 | 3 | 5 | 0 | 5 | 0 |
| clean_adm2 | 0 | 1.00 | 3 | 14 | 0 | 35 | 0 |
| clean_adm3 | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| clean_adm4 | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| installer | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| management_clean | 1573 | 0.67 | 5 | 37 | 0 | 7 | 0 |
| status_clean | 0 | 1.00 | 9 | 32 | 0 | 7 | 0 |
| pay | 0 | 1.00 | 2 | 39 | 0 | 7 | 0 |
| fecal_coliform_presence | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| subjective_quality | 0 | 1.00 | 18 | 20 | 0 | 4 | 0 |
| activity_id | 4757 | 0.00 | 36 | 36 | 0 | 3 | 0 |
| scheme_id | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| wpdx_id | 0 | 1.00 | 12 | 12 | 0 | 4760 | 0 |
| notes | 0 | 1.00 | 2 | 96 | 0 | 3502 | 0 |
| orig_lnk | 4757 | 0.00 | 84 | 84 | 0 | 1 | 0 |
| photo_lnk | 41 | 0.99 | 84 | 84 | 0 | 4719 | 0 |
| country_id | 0 | 1.00 | 2 | 2 | 0 | 1 | 0 |
| data_lnk | 0 | 1.00 | 79 | 96 | 0 | 2 | 0 |
| water_point_history | 0 | 1.00 | 142 | 834 | 0 | 4750 | 0 |
| clean_country_id | 0 | 1.00 | 3 | 3 | 0 | 1 | 0 |
| country_name | 0 | 1.00 | 7 | 7 | 0 | 1 | 0 |
| water_source | 0 | 1.00 | 8 | 30 | 0 | 4 | 0 |
| water_tech | 0 | 1.00 | 5 | 37 | 0 | 20 | 0 |
| adm2 | 0 | 1.00 | 3 | 14 | 0 | 33 | 0 |
| adm3 | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| management | 1573 | 0.67 | 5 | 47 | 0 | 7 | 0 |
| adm1 | 0 | 1.00 | 4 | 5 | 0 | 4 | 0 |
| New Georeferenced Column | 0 | 1.00 | 16 | 35 | 0 | 4760 | 0 |
| lat_lon_deg | 0 | 1.00 | 13 | 32 | 0 | 4760 | 0 |
| public_data_source | 0 | 1.00 | 84 | 102 | 0 | 2 | 0 |
| converted | 0 | 1.00 | 53 | 53 | 0 | 1 | 0 |
| created_timestamp | 0 | 1.00 | 22 | 22 | 0 | 2 | 0 |
| updated_timestamp | 0 | 1.00 | 22 | 22 | 0 | 2 | 0 |
| Geometry | 0 | 1.00 | 33 | 37 | 0 | 4760 | 0 |
| ADM2_EN | 0 | 1.00 | 3 | 14 | 0 | 30 | 0 |
| ADM2_PCODE | 0 | 1.00 | 8 | 8 | 0 | 30 | 0 |
| ADM1_EN | 0 | 1.00 | 4 | 4 | 0 | 1 | 0 |
| ADM1_PCODE | 0 | 1.00 | 5 | 5 | 0 | 1 | 0 |
Variable type: logical
| skim_variable | n_missing | complete_rate | mean | count |
|---|---|---|---|---|
| rehab_year | 4760 | 0 | NaN | : |
| rehabilitator | 4760 | 0 | NaN | : |
| is_urban | 0 | 1 | 0.39 | FAL: 2884, TRU: 1876 |
| latest_record | 0 | 1 | 1.00 | TRU: 4760 |
| status | 0 | 1 | 0.56 | TRU: 2642, FAL: 2118 |
Variable type: numeric
| skim_variable | n_missing | complete_rate | mean | sd | p0 | p25 | p50 | p75 | p100 | hist |
|---|---|---|---|---|---|---|---|---|---|---|
| row_id | 0 | 1.00 | 68550.48 | 10216.94 | 49601.00 | 66874.75 | 68244.50 | 69562.25 | 471319.00 | โโโโโ |
| lat_deg | 0 | 1.00 | 7.68 | 0.22 | 7.06 | 7.51 | 7.71 | 7.88 | 8.06 | โโโโโ |
| lon_deg | 0 | 1.00 | 4.54 | 0.21 | 4.08 | 4.36 | 4.56 | 4.71 | 5.06 | โโโโโ |
| install_year | 1144 | 0.76 | 2008.63 | 6.04 | 1917.00 | 2006.00 | 2010.00 | 2013.00 | 2015.00 | โโโโโ |
| fecal_coliform_value | 4760 | 0.00 | NaN | NA | NA | NA | NA | NA | NA | |
| distance_to_primary_road | 0 | 1.00 | 5021.53 | 5648.34 | 0.01 | 719.36 | 2972.78 | 7314.73 | 26909.86 | โโโโโ |
| distance_to_secondary_road | 0 | 1.00 | 3750.47 | 3938.63 | 0.15 | 460.90 | 2554.25 | 5791.94 | 19559.48 | โโโโโ |
| distance_to_tertiary_road | 0 | 1.00 | 1259.28 | 1680.04 | 0.02 | 121.25 | 521.77 | 1834.42 | 10966.27 | โโโโโ |
| distance_to_city | 0 | 1.00 | 16663.99 | 10960.82 | 53.05 | 7930.75 | 15030.41 | 24255.75 | 47934.34 | โโโโโ |
| distance_to_town | 0 | 1.00 | 16726.59 | 12452.65 | 30.00 | 6876.92 | 12204.53 | 27739.46 | 44020.64 | โโ โโโ |
| rehab_priority | 2654 | 0.44 | 489.33 | 1658.81 | 0.00 | 7.00 | 91.50 | 376.25 | 29697.00 | โโโโโ |
| water_point_population | 4 | 1.00 | 513.58 | 1458.92 | 0.00 | 14.00 | 119.00 | 433.25 | 29697.00 | โโโโโ |
| local_population_1km | 4 | 1.00 | 2727.16 | 4189.46 | 0.00 | 176.00 | 1032.00 | 3717.00 | 36118.00 | โโโโโ |
| crucialness_score | 798 | 0.83 | 0.26 | 0.28 | 0.00 | 0.07 | 0.15 | 0.35 | 1.00 | โโโโโ |
| pressure_score | 798 | 0.83 | 1.46 | 4.16 | 0.00 | 0.12 | 0.41 | 1.24 | 93.69 | โโโโโ |
| usage_capacity | 0 | 1.00 | 560.74 | 338.46 | 300.00 | 300.00 | 300.00 | 1000.00 | 1000.00 | โโโโโ |
| days_since_report | 0 | 1.00 | 2692.69 | 41.92 | 1483.00 | 2688.00 | 2693.00 | 2700.00 | 4645.00 | โโโโโ |
| staleness_score | 0 | 1.00 | 42.80 | 0.58 | 23.13 | 42.70 | 42.79 | 42.86 | 62.66 | โโโโโ |
| location_id | 0 | 1.00 | 235865.49 | 6657.60 | 23741.00 | 230638.75 | 236199.50 | 240061.25 | 267454.00 | โโโโโ |
| cluster_size | 0 | 1.00 | 1.05 | 0.25 | 1.00 | 1.00 | 1.00 | 1.00 | 4.00 | โโโโโ |
| lat_deg_original | 4760 | 0.00 | NaN | NA | NA | NA | NA | NA | NA | |
| lon_deg_original | 4760 | 0.00 | NaN | NA | NA | NA | NA | NA | NA | |
| count | 0 | 1.00 | 1.00 | 0.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | โโโโโ |
As observed, 'water_point_population' and 'local_population_1km' have 4 missing values. We can easily remove these datapoints since the count is
We also observed that 'usage_capacity' is numerical. However, we know that there are only 2 possible values, 300 and 1000. Given the context of the variable, this is a categorical variable.
The code chunk below removes any records that have NA values and converts the data type of 'usage_capacity' to factor.
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osun_wp_sf_clean <- osun_wp_sf %>%
# Removes NAs
filter_at(vars(status,
distance_to_primary_road,
distance_to_secondary_road,
distance_to_tertiary_road,
distance_to_city,
distance_to_town,
water_point_population,
local_population_1km,
usage_capacity,
is_urban,
water_source_clean),
all_vars(!is.na(.))) %>%
# Changes usage_capacity to categorical value
mutate(usage_capacity = as.factor(usage_capacity))As shown in the screenshot below, our new clean sf object has 4 less records.
Correlation Analysis
The code chunk below extracts our independent variables and drops the geometry feature of our sf object.
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osun_wp <- osun_wp_sf_clean %>%
select(c(7,35:39, 42:43, 46:47, 57)) %>%
st_set_geometry(NULL)The code chunk below uses corrplot.mixed() to create the correlation graph.
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cluster_vars.cor = cor(
osun_wp[,2:7])
corrplot.mixed(cluster_vars.cor,
lower = "ellipse",
upper = "number",
tl.pos = "lt",
diag = "l",
tl.col = "black")
Fortunately, we have no correlated independent variables.
Logistic Regression Model (Take 1)
The function glm() is used to calibrate a logistic regression for the water point status.
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model <- glm(status ~ distance_to_primary_road +
distance_to_secondary_road +
distance_to_tertiary_road +
distance_to_city +
distance_to_town +
water_point_population +
local_population_1km +
usage_capacity +
is_urban +
water_source_clean,
data = osun_wp_sf_clean,
family = binomial(link = 'logit'))The code chunk below uses blr_regress() to create a report from the modeling result.
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blr_regress(model) Model Overview
------------------------------------------------------------------------
Data Set Resp Var Obs. Df. Model Df. Residual Convergence
------------------------------------------------------------------------
data status 4756 4755 4744 TRUE
------------------------------------------------------------------------
Response Summary
--------------------------------------------------------
Outcome Frequency Outcome Frequency
--------------------------------------------------------
0 2114 1 2642
--------------------------------------------------------
Maximum Likelihood Estimates
-----------------------------------------------------------------------------------------------
Parameter DF Estimate Std. Error z value Pr(>|z|)
-----------------------------------------------------------------------------------------------
(Intercept) 1 0.3887 0.1124 3.4588 5e-04
distance_to_primary_road 1 0.0000 0.0000 -0.7153 0.4744
distance_to_secondary_road 1 0.0000 0.0000 -0.5530 0.5802
distance_to_tertiary_road 1 1e-04 0.0000 4.6708 0.0000
distance_to_city 1 0.0000 0.0000 -4.7574 0.0000
distance_to_town 1 0.0000 0.0000 -4.9170 0.0000
water_point_population 1 -5e-04 0.0000 -11.3686 0.0000
local_population_1km 1 3e-04 0.0000 19.2953 0.0000
usage_capacity1000 1 -0.6230 0.0697 -8.9366 0.0000
is_urbanTRUE 1 -0.2971 0.0819 -3.6294 3e-04
water_source_cleanProtected Shallow Well 1 0.5040 0.0857 5.8783 0.0000
water_source_cleanProtected Spring 1 1.2882 0.4388 2.9359 0.0033
-----------------------------------------------------------------------------------------------
Association of Predicted Probabilities and Observed Responses
---------------------------------------------------------------
% Concordant 0.7347 Somers' D 0.4693
% Discordant 0.2653 Gamma 0.4693
% Tied 0.0000 Tau-a 0.2318
Pairs 5585188 c 0.7347
---------------------------------------------------------------From the report above, we find that given a confidence of 95%, the variables 'distance_to primary_road' and 'distance_to_secondary_road' are not statistically significant since their p-values are above 0.05. For now, we will keep it in our model.
The code chunk below creates a confusion matrix using the function blr_confusion_matrix().
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blr_confusion_matrix(model, cutoff = 0.5)Confusion Matrix and Statistics
Reference
Prediction FALSE TRUE
0 1301 738
1 813 1904
Accuracy : 0.6739
No Information Rate : 0.4445
Kappa : 0.3373
McNemars's Test P-Value : 0.0602
Sensitivity : 0.7207
Specificity : 0.6154
Pos Pred Value : 0.7008
Neg Pred Value : 0.6381
Prevalence : 0.5555
Detection Rate : 0.4003
Detection Prevalence : 0.5713
Balanced Accuracy : 0.6680
Precision : 0.7008
Recall : 0.7207
'Positive' Class : 1
๐ LECTURE REVIEW!
As observed from the results above, our modelโs accuracy is 67.39%, sensitivity is 72.07% and specificity is 61.54 %. Our model can flag True Positives better than True Negatives.
Can we make a better model?
Geographically Weighted Logistic Regression Model (Take 1)
Convert sf to sp
Because gwmodel can only perform functions on sp objects, we need to convert our sf object using as_Spatial().
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osun_wp_sp <- osun_wp_sf_clean %>%
select(c(status,
distance_to_primary_road,
distance_to_secondary_road,
distance_to_tertiary_road,
distance_to_city,
distance_to_town,
water_point_population,
local_population_1km,
is_urban,
usage_capacity,
water_source_clean)) %>%
as_Spatial()
osun_wp_spclass : SpatialPointsDataFrame
features : 4756
extent : 182502.4, 290751, 340054.1, 450905.3 (xmin, xmax, ymin, ymax)
crs : +proj=tmerc +lat_0=4 +lon_0=8.5 +k=0.99975 +x_0=670553.98 +y_0=0 +a=6378249.145 +rf=293.465 +towgs84=-92,-93,122,0,0,0,0 +units=m +no_defs
variables : 11
names : status, distance_to_primary_road, distance_to_secondary_road, distance_to_tertiary_road, distance_to_city, distance_to_town, water_point_population, local_population_1km, is_urban, usage_capacity, water_source_clean
min values : 0, 0.014461356813335, 0.152195902540837, 0.017815121653488, 53.0461399623541, 30.0019777713073, 0, 0, 0, 1000, Borehole
max values : 1, 26909.8616132094, 19559.4793799085, 10966.2705628969, 47934.343603562, 44020.6393368124, 29697, 36118, 1, 300, Protected Spring Building Fixed Bandwidth
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bw.fixed <- bw.ggwr(status ~
distance_to_primary_road +
distance_to_secondary_road +
distance_to_tertiary_road +
distance_to_city +
distance_to_town +
water_point_population +
local_population_1km +
usage_capacity +
is_urban +
water_source_clean,
data = osun_wp_sp,
family = "binomial",
approach = "AIC",
kernel = "gaussian",
adaptive = FALSE,
longlat = FALSE)Take a cup of tea and have a break, it will take a few minutes.
-----A kind suggestion from GWmodel development group
Iteration Log-Likelihood:(With bandwidth: 95768.67 )
=========================
0 -2889
1 -2836
2 -2830
3 -2829
4 -2829
5 -2829
Fixed bandwidth: 95768.67 AICc value: 5684.357
Iteration Log-Likelihood:(With bandwidth: 59200.13 )
=========================
0 -2875
1 -2818
2 -2810
3 -2808
4 -2808
5 -2808
Fixed bandwidth: 59200.13 AICc value: 5646.785
Iteration Log-Likelihood:(With bandwidth: 36599.53 )
=========================
0 -2847
1 -2781
2 -2768
3 -2765
4 -2765
5 -2765
6 -2765
Fixed bandwidth: 36599.53 AICc value: 5575.148
Iteration Log-Likelihood:(With bandwidth: 22631.59 )
=========================
0 -2798
1 -2719
2 -2698
3 -2693
4 -2693
5 -2693
6 -2693
Fixed bandwidth: 22631.59 AICc value: 5466.883
Iteration Log-Likelihood:(With bandwidth: 13998.93 )
=========================
0 -2720
1 -2622
2 -2590
3 -2581
4 -2580
5 -2580
6 -2580
7 -2580
Fixed bandwidth: 13998.93 AICc value: 5324.578
Iteration Log-Likelihood:(With bandwidth: 8663.649 )
=========================
0 -2601
1 -2476
2 -2431
3 -2419
4 -2417
5 -2417
6 -2417
7 -2417
Fixed bandwidth: 8663.649 AICc value: 5163.61
Iteration Log-Likelihood:(With bandwidth: 5366.266 )
=========================
0 -2436
1 -2268
2 -2194
3 -2167
4 -2161
5 -2161
6 -2161
7 -2161
8 -2161
9 -2161
Fixed bandwidth: 5366.266 AICc value: 4990.587
Iteration Log-Likelihood:(With bandwidth: 3328.371 )
=========================
0 -2157
1 -1922
2 -1802
3 -1739
4 -1713
5 -1713
Fixed bandwidth: 3328.371 AICc value: 4798.288
Iteration Log-Likelihood:(With bandwidth: 2068.882 )
=========================
0 -1751
1 -1421
2 -1238
3 -1133
4 -1084
5 -1084
Fixed bandwidth: 2068.882 AICc value: 4837.017
Iteration Log-Likelihood:(With bandwidth: 4106.777 )
=========================
0 -2297
1 -2095
2 -1997
3 -1951
4 -1938
5 -1936
6 -1936
7 -1936
8 -1936
Fixed bandwidth: 4106.777 AICc value: 4873.161
Iteration Log-Likelihood:(With bandwidth: 2847.289 )
=========================
0 -2036
1 -1771
2 -1633
3 -1558
4 -1525
5 -1525
Fixed bandwidth: 2847.289 AICc value: 4768.192
Iteration Log-Likelihood:(With bandwidth: 2549.964 )
=========================
0 -1941
1 -1655
2 -1503
3 -1417
4 -1378
5 -1378
Fixed bandwidth: 2549.964 AICc value: 4762.212
Iteration Log-Likelihood:(With bandwidth: 2366.207 )
=========================
0 -1874
1 -1573
2 -1410
3 -1316
4 -1274
5 -1274
Fixed bandwidth: 2366.207 AICc value: 4773.081
Iteration Log-Likelihood:(With bandwidth: 2663.532 )
=========================
0 -1979
1 -1702
2 -1555
3 -1474
4 -1438
5 -1438
Fixed bandwidth: 2663.532 AICc value: 4762.568
Iteration Log-Likelihood:(With bandwidth: 2479.775 )
=========================
0 -1917
1 -1625
2 -1468
3 -1380
4 -1339
5 -1339
Fixed bandwidth: 2479.775 AICc value: 4764.294
Iteration Log-Likelihood:(With bandwidth: 2593.343 )
=========================
0 -1956
1 -1674
2 -1523
3 -1439
4 -1401
5 -1401
Fixed bandwidth: 2593.343 AICc value: 4761.813
Iteration Log-Likelihood:(With bandwidth: 2620.153 )
=========================
0 -1965
1 -1685
2 -1536
3 -1453
4 -1415
5 -1415
Fixed bandwidth: 2620.153 AICc value: 4761.89
Iteration Log-Likelihood:(With bandwidth: 2576.774 )
=========================
0 -1950
1 -1667
2 -1515
3 -1431
4 -1393
5 -1393
Fixed bandwidth: 2576.774 AICc value: 4761.889
Iteration Log-Likelihood:(With bandwidth: 2603.584 )
=========================
0 -1960
1 -1678
2 -1528
3 -1445
4 -1407
5 -1407
Fixed bandwidth: 2603.584 AICc value: 4761.813
Iteration Log-Likelihood:(With bandwidth: 2609.913 )
=========================
0 -1962
1 -1680
2 -1531
3 -1448
4 -1410
5 -1410
Fixed bandwidth: 2609.913 AICc value: 4761.831
Iteration Log-Likelihood:(With bandwidth: 2599.672 )
=========================
0 -1958
1 -1676
2 -1526
3 -1443
4 -1405
5 -1405
Fixed bandwidth: 2599.672 AICc value: 4761.809
Iteration Log-Likelihood:(With bandwidth: 2597.255 )
=========================
0 -1957
1 -1675
2 -1525
3 -1441
4 -1403
5 -1403
Fixed bandwidth: 2597.255 AICc value: 4761.809 Press to toggle code
# since our data is projected, set longlat = FALSEPress to toggle code
bw.fixed[1] 2599.672The result is 2599.67 km.
Press to toggle code
gwlr.fixed <- ggwr.basic(status ~
distance_to_primary_road +
distance_to_secondary_road +
distance_to_tertiary_road +
distance_to_city +
distance_to_town +
water_point_population +
local_population_1km +
usage_capacity +
is_urban +
water_source_clean,
data = osun_wp_sp,
bw = bw.fixed,
family = "binomial",
kernel = "gaussian",
adaptive = FALSE,
longlat = FALSE) Iteration Log-Likelihood
=========================
0 -1958
1 -1676
2 -1526
3 -1443
4 -1405
5 -1405 Press to toggle code
gwlr.fixed ***********************************************************************
* Package GWmodel *
***********************************************************************
Program starts at: 2022-12-17 23:24:00
Call:
ggwr.basic(formula = status ~ distance_to_primary_road + distance_to_secondary_road +
distance_to_tertiary_road + distance_to_city + distance_to_town +
water_point_population + local_population_1km + usage_capacity +
is_urban + water_source_clean, data = osun_wp_sp, bw = bw.fixed,
family = "binomial", kernel = "gaussian", adaptive = FALSE,
longlat = FALSE)
Dependent (y) variable: status
Independent variables: distance_to_primary_road distance_to_secondary_road distance_to_tertiary_road distance_to_city distance_to_town water_point_population local_population_1km usage_capacity is_urban water_source_clean
Number of data points: 4756
Used family: binomial
***********************************************************************
* Results of Generalized linear Regression *
***********************************************************************
Call:
NULL
Deviance Residuals:
Min 1Q Median 3Q Max
-124.555 -1.755 1.072 1.742 34.333
Coefficients:
Estimate Std. Error z value Pr(>|z|)
Intercept 3.887e-01 1.124e-01 3.459 0.000543
distance_to_primary_road -4.642e-06 6.490e-06 -0.715 0.474422
distance_to_secondary_road -5.143e-06 9.299e-06 -0.553 0.580230
distance_to_tertiary_road 9.683e-05 2.073e-05 4.671 3.00e-06
distance_to_city -1.686e-05 3.544e-06 -4.757 1.96e-06
distance_to_town -1.480e-05 3.009e-06 -4.917 8.79e-07
water_point_population -5.097e-04 4.484e-05 -11.369 < 2e-16
local_population_1km 3.451e-04 1.788e-05 19.295 < 2e-16
usage_capacity1000 -6.230e-01 6.972e-02 -8.937 < 2e-16
is_urbanTRUE -2.971e-01 8.185e-02 -3.629 0.000284
water_source_cleanProtected Shallow Well 5.040e-01 8.574e-02 5.878 4.14e-09
water_source_cleanProtected Spring 1.288e+00 4.388e-01 2.936 0.003325
Intercept ***
distance_to_primary_road
distance_to_secondary_road
distance_to_tertiary_road ***
distance_to_city ***
distance_to_town ***
water_point_population ***
local_population_1km ***
usage_capacity1000 ***
is_urbanTRUE ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 6534.5 on 4755 degrees of freedom
Residual deviance: 5688.0 on 4744 degrees of freedom
AIC: 5712
Number of Fisher Scoring iterations: 5
AICc: 5712.099
Pseudo R-square value: 0.1295351
***********************************************************************
* Results of Geographically Weighted Regression *
***********************************************************************
*********************Model calibration information*********************
Kernel function: gaussian
Fixed bandwidth: 2599.672
Regression points: the same locations as observations are used.
Distance metric: A distance matrix is specified for this model calibration.
************Summary of Generalized GWR coefficient estimates:**********
Min. 1st Qu. Median
Intercept -8.7228e+02 -4.9955e+00 1.7600e+00
distance_to_primary_road -1.9389e-02 -4.8031e-04 2.9618e-05
distance_to_secondary_road -1.5921e-02 -3.7551e-04 1.2317e-04
distance_to_tertiary_road -1.5618e-02 -4.2368e-04 7.6179e-05
distance_to_city -1.8416e-02 -5.6217e-04 -1.2726e-04
distance_to_town -2.2411e-02 -5.7283e-04 -1.5155e-04
water_point_population -5.2208e-02 -2.2767e-03 -9.8875e-04
local_population_1km -1.2698e-01 4.9952e-04 1.0638e-03
usage_capacity1000 -2.0772e+01 -9.7231e-01 -4.1592e-01
is_urbanTRUE -1.9790e+02 -4.2908e+00 -1.6864e+00
water_source_cleanProtected.Shallow.Well -2.0789e+01 -4.5190e-01 5.3340e-01
water_source_cleanProtected.Spring -5.2235e+02 -5.5977e+00 2.5441e+00
3rd Qu. Max.
Intercept 1.2763e+01 1073.2154
distance_to_primary_road 4.8443e-04 0.0142
distance_to_secondary_road 6.0692e-04 0.0258
distance_to_tertiary_road 6.6814e-04 0.0128
distance_to_city 2.3718e-04 0.0150
distance_to_town 1.9271e-04 0.0224
water_point_population 5.0102e-04 0.1309
local_population_1km 1.8157e-03 0.0392
usage_capacity1000 3.0322e-01 5.9281
is_urbanTRUE 1.2841e+00 744.3097
water_source_cleanProtected.Shallow.Well 1.7849e+00 67.6343
water_source_cleanProtected.Spring 6.7663e+00 317.4123
************************Diagnostic information*************************
Number of data points: 4756
GW Deviance: 2795.084
AIC : 4414.606
AICc : 4747.423
Pseudo R-square value: 0.5722559
***********************************************************************
Program stops at: 2022-12-17 23:24:32 As observed from above, the AIC values dropped down from 5712.10 to 4414.61.
Converting SDF into sf data.frame
Press to toggle code
gwr.fixed <- as.data.frame(gwlr.fixed$SDF)Since 'yhat' is probability, we need to create a new logical column called 'most' to label as 1 or 0 if the 'yhat' is greater or equal to 0.5.
Press to toggle code
gwr.fixed <- gwr.fixed %>%
mutate(most = ifelse(
gwr.fixed$yhat >= 0.5, T, F))The code chunk below creates the confustion matrix of the model.
Press to toggle code
gwr.fixed$y <- as.factor(gwr.fixed$y)
gwr.fixed$most <- as.factor(gwr.fixed$most)
CM <- confusionMatrix(data = gwr.fixed$most, reference = gwr.fixed$y)
CMConfusion Matrix and Statistics
Reference
Prediction FALSE TRUE
FALSE 1824 263
TRUE 290 2379
Accuracy : 0.8837
95% CI : (0.8743, 0.8927)
No Information Rate : 0.5555
P-Value [Acc > NIR] : <2e-16
Kappa : 0.7642
Mcnemar's Test P-Value : 0.2689
Sensitivity : 0.8628
Specificity : 0.9005
Pos Pred Value : 0.8740
Neg Pred Value : 0.8913
Prevalence : 0.4445
Detection Rate : 0.3835
Detection Prevalence : 0.4388
Balanced Accuracy : 0.8816
'Positive' Class : FALSE
Press to toggle code
osun_wp_sf_selected <- osun_wp_sf_clean %>%
select(c(ADM2_EN, ADM2_PCODE,
ADM1_EN, ADM1_PCODE,
status))The logistic regression model has accuracy of 88.37%, sensitivity of 86.28%, and specificity of 90.05%.
The code chunk below combines our water point data and model.
Press to toggle code
gwr_sf.fixed <- cbind(osun_wp_sf_selected, gwr.fixed)Visualizing Coefficient Estimates
The code chunk below creates an interactive point map of the probability.
Press to toggle code
tmap_mode("view")
prob_T <- tm_shape(osun) +
tm_polygons(alpha = 0.1) +
tm_shape(gwr_sf.fixed) +
tm_dots(col = "yhat",
palette = "RdPu",
border.col = "#704276",
border.lwd = 1) +
tm_view(set.zoom.limits = c(8,14))
prob_TLogistic Regression Model (Take 2)
Now, letโs try making the model without the two variables that were not significant 'distance_to_primary_road' and 'distance_to_secondary_road'.
Press to toggle code
model2 <- glm(status ~ distance_to_tertiary_road +
distance_to_city +
distance_to_town +
water_point_population +
local_population_1km +
usage_capacity +
is_urban +
water_source_clean,
data = osun_wp_sf_clean,
family = binomial(link = 'logit'))The code chunk below uses blr_regress() to create a report from the modeling result.
Press to toggle code
blr_regress(model2) Model Overview
------------------------------------------------------------------------
Data Set Resp Var Obs. Df. Model Df. Residual Convergence
------------------------------------------------------------------------
data status 4756 4755 4746 TRUE
------------------------------------------------------------------------
Response Summary
--------------------------------------------------------
Outcome Frequency Outcome Frequency
--------------------------------------------------------
0 2114 1 2642
--------------------------------------------------------
Maximum Likelihood Estimates
-----------------------------------------------------------------------------------------------
Parameter DF Estimate Std. Error z value Pr(>|z|)
-----------------------------------------------------------------------------------------------
(Intercept) 1 0.3540 0.1055 3.3541 8e-04
distance_to_tertiary_road 1 1e-04 0.0000 4.9096 0.0000
distance_to_city 1 0.0000 0.0000 -5.2022 0.0000
distance_to_town 1 0.0000 0.0000 -5.4660 0.0000
water_point_population 1 -5e-04 0.0000 -11.3902 0.0000
local_population_1km 1 3e-04 0.0000 19.4069 0.0000
usage_capacity1000 1 -0.6206 0.0697 -8.9081 0.0000
is_urbanTRUE 1 -0.2667 0.0747 -3.5690 4e-04
water_source_cleanProtected Shallow Well 1 0.4947 0.0850 5.8228 0.0000
water_source_cleanProtected Spring 1 1.2790 0.4384 2.9174 0.0035
-----------------------------------------------------------------------------------------------
Association of Predicted Probabilities and Observed Responses
---------------------------------------------------------------
% Concordant 0.7349 Somers' D 0.4697
% Discordant 0.2651 Gamma 0.4697
% Tied 0.0000 Tau-a 0.2320
Pairs 5585188 c 0.7349
---------------------------------------------------------------All variables are now significant.
The code chunk below creates a confusion matrix using the function blr_confusion_matrix().
Press to toggle code
blr_confusion_matrix(model2, cutoff = 0.5)Confusion Matrix and Statistics
Reference
Prediction FALSE TRUE
0 1300 743
1 814 1899
Accuracy : 0.6726
No Information Rate : 0.4445
Kappa : 0.3348
McNemars's Test P-Value : 0.0761
Sensitivity : 0.7188
Specificity : 0.6149
Pos Pred Value : 0.7000
Neg Pred Value : 0.6363
Prevalence : 0.5555
Detection Rate : 0.3993
Detection Prevalence : 0.5704
Balanced Accuracy : 0.6669
Precision : 0.7000
Recall : 0.7188
'Positive' Class : 1Despite removing the two variables, the accuracy, sensitivity and specificity did not improve and did not change much. The small change in its value actually got worse by a few decimal values. Accuracy dropped by 0.13%, sensitivity dropped by 0.19% and specificity dropped by 0.05%.
Geographically Weighted Logistic Regression Model (Take 2)
Convert sf to sp
Because gwmodel can only perform functions on sp objects, we need to convert our sf object using as_Spatial() but without the two not statistically significant variables.
Press to toggle code
osun_wp_sp2 <- osun_wp_sf_clean %>%
select(c(status,
distance_to_tertiary_road,
distance_to_city,
distance_to_town,
water_point_population,
local_population_1km,
is_urban,
usage_capacity,
water_source_clean)) %>%
as_Spatial()
osun_wp_sp2class : SpatialPointsDataFrame
features : 4756
extent : 182502.4, 290751, 340054.1, 450905.3 (xmin, xmax, ymin, ymax)
crs : +proj=tmerc +lat_0=4 +lon_0=8.5 +k=0.99975 +x_0=670553.98 +y_0=0 +a=6378249.145 +rf=293.465 +towgs84=-92,-93,122,0,0,0,0 +units=m +no_defs
variables : 9
names : status, distance_to_tertiary_road, distance_to_city, distance_to_town, water_point_population, local_population_1km, is_urban, usage_capacity, water_source_clean
min values : 0, 0.017815121653488, 53.0461399623541, 30.0019777713073, 0, 0, 0, 1000, Borehole
max values : 1, 10966.2705628969, 47934.343603562, 44020.6393368124, 29697, 36118, 1, 300, Protected Spring Building Fixed Bandwidth
Press to toggle code
bw.fixed2 <- bw.ggwr(status ~
distance_to_tertiary_road +
distance_to_city +
distance_to_town +
water_point_population +
local_population_1km +
usage_capacity +
is_urban +
water_source_clean,
data = osun_wp_sp2,
family = "binomial",
approach = "AIC",
kernel = "gaussian",
adaptive = FALSE,
longlat = FALSE)Take a cup of tea and have a break, it will take a few minutes.
-----A kind suggestion from GWmodel development group
Iteration Log-Likelihood:(With bandwidth: 95768.67 )
=========================
0 -2890
1 -2837
2 -2830
3 -2829
4 -2829
5 -2829
Fixed bandwidth: 95768.67 AICc value: 5681.18
Iteration Log-Likelihood:(With bandwidth: 59200.13 )
=========================
0 -2878
1 -2820
2 -2812
3 -2810
4 -2810
5 -2810
Fixed bandwidth: 59200.13 AICc value: 5645.901
Iteration Log-Likelihood:(With bandwidth: 36599.53 )
=========================
0 -2854
1 -2790
2 -2777
3 -2774
4 -2774
5 -2774
6 -2774
Fixed bandwidth: 36599.53 AICc value: 5585.354
Iteration Log-Likelihood:(With bandwidth: 22631.59 )
=========================
0 -2810
1 -2732
2 -2711
3 -2707
4 -2707
5 -2707
6 -2707
Fixed bandwidth: 22631.59 AICc value: 5481.877
Iteration Log-Likelihood:(With bandwidth: 13998.93 )
=========================
0 -2732
1 -2635
2 -2604
3 -2597
4 -2596
5 -2596
6 -2596
Fixed bandwidth: 13998.93 AICc value: 5333.718
Iteration Log-Likelihood:(With bandwidth: 8663.649 )
=========================
0 -2624
1 -2502
2 -2459
3 -2447
4 -2446
5 -2446
6 -2446
7 -2446
Fixed bandwidth: 8663.649 AICc value: 5178.493
Iteration Log-Likelihood:(With bandwidth: 5366.266 )
=========================
0 -2478
1 -2319
2 -2250
3 -2225
4 -2219
5 -2219
6 -2220
7 -2220
8 -2220
9 -2220
Fixed bandwidth: 5366.266 AICc value: 5022.016
Iteration Log-Likelihood:(With bandwidth: 3328.371 )
=========================
0 -2222
1 -2002
2 -1894
3 -1838
4 -1818
5 -1814
6 -1814
Fixed bandwidth: 3328.371 AICc value: 4827.587
Iteration Log-Likelihood:(With bandwidth: 2068.882 )
=========================
0 -1837
1 -1528
2 -1357
3 -1261
4 -1222
5 -1222
Fixed bandwidth: 2068.882 AICc value: 4772.046
Iteration Log-Likelihood:(With bandwidth: 1290.476 )
=========================
0 -1403
1 -1016
2 -807.3
3 -680.2
4 -680.2
Fixed bandwidth: 1290.476 AICc value: 5809.721
Iteration Log-Likelihood:(With bandwidth: 2549.964 )
=========================
0 -2019
1 -1753
2 -1614
3 -1538
4 -1506
5 -1506
Fixed bandwidth: 2549.964 AICc value: 4764.056
Iteration Log-Likelihood:(With bandwidth: 2847.289 )
=========================
0 -2108
1 -1862
2 -1736
3 -1670
4 -1644
5 -1644
Fixed bandwidth: 2847.289 AICc value: 4791.834
Iteration Log-Likelihood:(With bandwidth: 2366.207 )
=========================
0 -1955
1 -1675
2 -1525
3 -1441
4 -1407
5 -1407
Fixed bandwidth: 2366.207 AICc value: 4755.524
Iteration Log-Likelihood:(With bandwidth: 2252.639 )
=========================
0 -1913
1 -1623
2 -1465
3 -1376
4 -1341
5 -1341
Fixed bandwidth: 2252.639 AICc value: 4759.188
Iteration Log-Likelihood:(With bandwidth: 2436.396 )
=========================
0 -1980
1 -1706
2 -1560
3 -1479
4 -1446
5 -1446
Fixed bandwidth: 2436.396 AICc value: 4756.675
Iteration Log-Likelihood:(With bandwidth: 2322.828 )
=========================
0 -1940
1 -1656
2 -1503
3 -1417
4 -1382
5 -1382
Fixed bandwidth: 2322.828 AICc value: 4756.471
Iteration Log-Likelihood:(With bandwidth: 2393.017 )
=========================
0 -1965
1 -1687
2 -1539
3 -1456
4 -1422
5 -1422
Fixed bandwidth: 2393.017 AICc value: 4755.57
Iteration Log-Likelihood:(With bandwidth: 2349.638 )
=========================
0 -1949
1 -1668
2 -1517
3 -1432
4 -1398
5 -1398
Fixed bandwidth: 2349.638 AICc value: 4755.753
Iteration Log-Likelihood:(With bandwidth: 2376.448 )
=========================
0 -1959
1 -1680
2 -1530
3 -1447
4 -1413
5 -1413
Fixed bandwidth: 2376.448 AICc value: 4755.48
Iteration Log-Likelihood:(With bandwidth: 2382.777 )
=========================
0 -1961
1 -1683
2 -1534
3 -1450
4 -1416
5 -1416
Fixed bandwidth: 2382.777 AICc value: 4755.491
Iteration Log-Likelihood:(With bandwidth: 2372.536 )
=========================
0 -1958
1 -1678
2 -1528
3 -1445
4 -1411
5 -1411
Fixed bandwidth: 2372.536 AICc value: 4755.488
Iteration Log-Likelihood:(With bandwidth: 2378.865 )
=========================
0 -1960
1 -1681
2 -1532
3 -1448
4 -1414
5 -1414
Fixed bandwidth: 2378.865 AICc value: 4755.481
Iteration Log-Likelihood:(With bandwidth: 2374.954 )
=========================
0 -1959
1 -1679
2 -1530
3 -1446
4 -1412
5 -1412
Fixed bandwidth: 2374.954 AICc value: 4755.482
Iteration Log-Likelihood:(With bandwidth: 2377.371 )
=========================
0 -1959
1 -1680
2 -1531
3 -1447
4 -1413
5 -1413
Fixed bandwidth: 2377.371 AICc value: 4755.48
Iteration Log-Likelihood:(With bandwidth: 2377.942 )
=========================
0 -1960
1 -1680
2 -1531
3 -1448
4 -1414
5 -1414
Fixed bandwidth: 2377.942 AICc value: 4755.48
Iteration Log-Likelihood:(With bandwidth: 2377.018 )
=========================
0 -1959
1 -1680
2 -1531
3 -1447
4 -1413
5 -1413
Fixed bandwidth: 2377.018 AICc value: 4755.48 Press to toggle code
# since our data is projected, set longlat = FALSEPress to toggle code
bw.fixed2[1] 2377.371The result is 2377.371 km.
Press to toggle code
gwlr.fixed2 <- ggwr.basic(status ~
distance_to_tertiary_road +
distance_to_city +
distance_to_town +
water_point_population +
local_population_1km +
usage_capacity +
is_urban +
water_source_clean,
data = osun_wp_sp2,
bw = bw.fixed2,
family = "binomial",
kernel = "gaussian",
adaptive = FALSE,
longlat = FALSE) Iteration Log-Likelihood
=========================
0 -1959
1 -1680
2 -1531
3 -1447
4 -1413
5 -1413 Press to toggle code
gwlr.fixed2 ***********************************************************************
* Package GWmodel *
***********************************************************************
Program starts at: 2022-12-17 23:34:15
Call:
ggwr.basic(formula = status ~ distance_to_tertiary_road + distance_to_city +
distance_to_town + water_point_population + local_population_1km +
usage_capacity + is_urban + water_source_clean, data = osun_wp_sp2,
bw = bw.fixed2, family = "binomial", kernel = "gaussian",
adaptive = FALSE, longlat = FALSE)
Dependent (y) variable: status
Independent variables: distance_to_tertiary_road distance_to_city distance_to_town water_point_population local_population_1km usage_capacity is_urban water_source_clean
Number of data points: 4756
Used family: binomial
***********************************************************************
* Results of Generalized linear Regression *
***********************************************************************
Call:
NULL
Deviance Residuals:
Min 1Q Median 3Q Max
-129.368 -1.750 1.074 1.742 34.126
Coefficients:
Estimate Std. Error z value Pr(>|z|)
Intercept 3.540e-01 1.055e-01 3.354 0.000796
distance_to_tertiary_road 1.001e-04 2.040e-05 4.910 9.13e-07
distance_to_city -1.764e-05 3.391e-06 -5.202 1.97e-07
distance_to_town -1.544e-05 2.825e-06 -5.466 4.60e-08
water_point_population -5.098e-04 4.476e-05 -11.390 < 2e-16
local_population_1km 3.452e-04 1.779e-05 19.407 < 2e-16
usage_capacity1000 -6.206e-01 6.966e-02 -8.908 < 2e-16
is_urbanTRUE -2.667e-01 7.474e-02 -3.569 0.000358
water_source_cleanProtected Shallow Well 4.947e-01 8.496e-02 5.823 5.79e-09
water_source_cleanProtected Spring 1.279e+00 4.384e-01 2.917 0.003530
Intercept ***
distance_to_tertiary_road ***
distance_to_city ***
distance_to_town ***
water_point_population ***
local_population_1km ***
usage_capacity1000 ***
is_urbanTRUE ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 6534.5 on 4755 degrees of freedom
Residual deviance: 5688.9 on 4746 degrees of freedom
AIC: 5708.9
Number of Fisher Scoring iterations: 5
AICc: 5708.923
Pseudo R-square value: 0.129406
***********************************************************************
* Results of Geographically Weighted Regression *
***********************************************************************
*********************Model calibration information*********************
Kernel function: gaussian
Fixed bandwidth: 2377.371
Regression points: the same locations as observations are used.
Distance metric: A distance matrix is specified for this model calibration.
************Summary of Generalized GWR coefficient estimates:**********
Min. 1st Qu. Median
Intercept -3.7021e+02 -4.3797e+00 3.5590e+00
distance_to_tertiary_road -3.1622e-02 -4.5462e-04 9.1291e-05
distance_to_city -5.4555e-02 -6.5623e-04 -1.3507e-04
distance_to_town -8.6549e-03 -5.2754e-04 -1.6785e-04
water_point_population -2.9696e-02 -2.2705e-03 -1.2277e-03
local_population_1km -7.7730e-02 4.4281e-04 1.0548e-03
usage_capacity1000 -5.5889e+01 -1.0347e+00 -4.1960e-01
is_urbanTRUE -7.3554e+02 -3.4675e+00 -1.6596e+00
water_source_cleanProtected.Shallow.Well -1.8842e+02 -4.7295e-01 6.2378e-01
water_source_cleanProtected.Spring -1.3630e+03 -5.3436e+00 2.7714e+00
3rd Qu. Max.
Intercept 1.3755e+01 2171.6373
distance_to_tertiary_road 6.3011e-04 0.0237
distance_to_city 1.5921e-04 0.0162
distance_to_town 2.4490e-04 0.0179
water_point_population 4.5879e-04 0.0765
local_population_1km 1.8479e-03 0.0333
usage_capacity1000 3.9113e-01 9.2449
is_urbanTRUE 1.0554e+00 995.1840
water_source_cleanProtected.Shallow.Well 1.9564e+00 66.8914
water_source_cleanProtected.Spring 7.0805e+00 208.3749
************************Diagnostic information*************************
Number of data points: 4756
GW Deviance: 2815.659
AIC : 4418.776
AICc : 4744.213
Pseudo R-square value: 0.5691072
***********************************************************************
Program stops at: 2022-12-17 23:34:44 As observed from above, the AIC values dropped down from 5708.923 to 4418.776 which is still higher than the that of the first geographically weighted logistic regression with AIC at 4414.61.
Converting SDF into sf data.frame
Press to toggle code
gwr.fixed2 <- as.data.frame(gwlr.fixed2$SDF)Since 'yhat' is probability, we need to create a new logical column called 'most' to label as 1 or 0 if the 'yhat' is greater or equal to 0.5.
Press to toggle code
gwr.fixed2 <- gwr.fixed2 %>%
mutate(most = ifelse(
gwr.fixed2$yhat >= 0.5, T, F))The code chunk below creates the confusion matrix of the model.
Press to toggle code
gwr.fixed2$y <- as.factor(gwr.fixed2$y)
gwr.fixed2$most <- as.factor(gwr.fixed2$most)
CM2 <- confusionMatrix(data = gwr.fixed2$most, reference = gwr.fixed2$y)
CM2Confusion Matrix and Statistics
Reference
Prediction FALSE TRUE
FALSE 1833 268
TRUE 281 2374
Accuracy : 0.8846
95% CI : (0.8751, 0.8935)
No Information Rate : 0.5555
P-Value [Acc > NIR] : <2e-16
Kappa : 0.7661
Mcnemar's Test P-Value : 0.6085
Sensitivity : 0.8671
Specificity : 0.8986
Pos Pred Value : 0.8724
Neg Pred Value : 0.8942
Prevalence : 0.4445
Detection Rate : 0.3854
Detection Prevalence : 0.4418
Balanced Accuracy : 0.8828
'Positive' Class : FALSE
The accuracy is 88.46%, the sensitivity is 86.71% and the specificity is 89.86%. While the accuracy increased by 0.09% and the sensitivity increased by 0.33%, the specificity decreased by 0.19%. The improvement is quite negligible, but there is more improvement here than the simple logistic model.
The code chunk below combines our water point data and model.
Press to toggle code
gwr_sf.fixed2 <- cbind(osun_wp_sf_selected, gwr.fixed2)Visualizing Coefficient Estimates
The code chunk below creates an interactive point map of the probability.
Press to toggle code
tmap_mode("view")
prob_T2 <- tm_shape(osun) +
tm_polygons(alpha = 0.1) +
tm_shape(gwr_sf.fixed2) +
tm_dots(col = "yhat",
palette = "RdPu",
border.col = "#704276",
border.lwd = 1) +
tm_view(set.zoom.limits = c(8,14))
prob_T2